# Generalizations of Prime Ideals of Semirings

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The additive reduct of a band semiring is a regular band.

- Generalized Derivations and Left Ideals in Prime and Semiprime Rings. Dhara, Basudeb; Pattanayak, Atanu // ISRN Algebra;2011, Special section p1
Let R be an associative ring, Î» a nonzero left ideal of R, d : R â†’ R a derivation and G : R â†’ R a generalized derivation. In this paper, we study the following situations in prime and semiprime rings: (1) G (x o y) = a(xy Â± yx); (2) G[x, y] = a(xy Â± yx); (3) d(x) o d(y) =...

- A Zariski Topology For Semimodules. Atani, Shahabaddin Ebrahimi; Atani, Reza Ebrahimi; Tekir, �nsal // European Journal of Pure & Applied Mathematics;2011, Vol. 4 Issue 3, p251
Given a very strong multiplication semimidule M over a commutative semiring R, a Zariski topology is defined on the spectrum Speck(M) of prime k-subsemimodules of M. The properties and possible structures of this topology are studied.

- On Exact Sequence of Semimodules over Semirings. Ninu Chaudhari, Jayprakash; Ravindra Bonde, Dipak // ISRN Algebra;2013, p1
We introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms and generalize some results of module theory to semimodules over semirings. Indeed, we prove, "If 0 â†’ Lf â†’ Mg â†’ N â†’ 0 is a split exact sequence of R-semimodules and...

- Weakly Special Radical Class and Special Radical Class of Ternary Semirings. Dutta, Tapan K.; Kar Ping Shum; Mandal, Shobhan // European Journal of Pure & Applied Mathematics;2012, Vol. 5 Issue 4, p401
In this paper, we consider the weakly special radical classes and special radical classes of ternary semirings. Some of our results are similar to those in rings theory as well as in semiring theory. In particular, the upper radicals of the above two classes are determined.

- A scheme over prime spectrum of modules. ABBASI, Ahmad; HASSANZADEH LELEKAMI, Dawood // Turkish Journal of Mathematics;Mar2013, Vol. 37 Issue 2, p202
Let R be a commutative ring with nonzero identity and let M be an R-module with X = Spec (M). It is introduced a scheme OX on the prime spectrum of M and some of its properties have been investigated.

- Modules Whose Certain Submodules Are Prime. Behboodi, M.; Karamzadeh, O. A. S.; Koohy, H. // Vietnam Journal of Mathematics;Sep2004, Vol. 32 Issue 3, p303
Modules in which every proper submodule (resp. proper nonzero submodule) is prime (called fully prime (almost fully prime)) and with some other related notions are fully investigated. It is shown that over a commutative ring R, an R-module M is fully prime (fully semiprime) if and only if M is a...

- KÃ¶the's upper nil radical for modules. Groenewald, N.; Ssevviiri, D. // Acta Mathematica Hungarica;Mar2013, Vol. 138 Issue 4, p295
Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$. It is shown that $\mathcal{S}(N)$ is equal to the...

- Almost Graded Prime Ideals. Jaber, Ameer; Bataineh, Malik; Khashan, Hani // Journal of Mathematics & Statistics;2008, Vol. 4 Issue 4, p232
Problem Statement: Graded commutative ring with unity over an abelian group were introduced by many authors such as T. Y. Lam and C. T. C. Wall, and almost prime ideals over commutative rings with unity were introduced by S.M. Batwadeker and P.K. Sharma, and this forced us to try to extend the...